Is the unit sphere in $\Bbb R^4$ path connected?

Question

I am asked whether the unit sphere $$X=\{(x,y,z,w)|x^2 + y^2 + z^2 + w^2 = 1 \}\subset \mathbb{R}^4,$$ is path connected or not.

I just know that $X$ is a closed subset. How can I answer this question?

Is there any hint? Thank you very much.

Answer 1

The set $X$ is the unit sphere in $\Bbb{R}^4$. Any two distinct $x,y\in\Bbb{R}^4$ are contained in some plane $Y\subset\Bbb{R}^4$, and hence $x,y\in X\cap Y$. But $X\cap Y$ is then a circle in $Y$, which is of course path connected. So $x$ and $y$ are connected by a path in $X\cap Y\subset X$, hence $X$ is path connected.